© H. Heck 2008Section 5.41 Module 5:Advanced Transmission Lines Topic 4: Frequency Domain Analysis OGI ECE564 Howard Heck

© H. Heck 2008 Section 5.4 1

Module 5: Advanced Transmission LinesTopic 4: Frequency Domain Analysis

OGI ECE564

Howard Heck


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Where Are We?

1. Introduction

2. Transmission Line Basics

3. Analysis Tools

4. Metrics & Methodology

5. Advanced Transmission Lines1. Losses

2. Intersymbol Interference

3. Crosstalk

4. Frequency Domain Analysis

5. 2 Port Networks & S-Parameters6. Multi-Gb/s Signaling

7. Special Topics


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Contents

Motivation Wave Equation Revisited Frequency Dependence Reflection Coefficient and Impedance Input Impedance Examples Summary References


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Motivation

At high frequencies, losses become significant. This makes time domain analysis difficult, as the properties are frequency dependent. Skin effect, dielectric loss & dispersion

We need to develop the means to understand those effects. Example: How would we measure R, L, G, C for a PCB trace?

Frequency domain analysis allows discrete characterization of a linear network at each frequency. Characterization at a single frequency is much easier

Frequency Analysis has advantages: Ease and accuracy of measurement at high frequencies Simplified mathematics Allows separation of electrical phenomena (loss, resonance …

etc).


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Key Concepts

The input impedance & the input reflection coefficient of a transmission line is dependent on: Termination and characteristic impedance Delay Frequency

S-Parameters are used to extract electrical parameters. Transmission line parameters (R,L,C,G, TD and Zo) Vias, connectors, socket … equivalent circuits

Periodic behavior of S-parameters can be used to gain intuition of signal integrity problems.

We’ll study S-parameters in section 5.5.


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Derive the lossy wave equation Add a sinusoidal stimulus


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Wave Equation Revisited

Goal: derive the frequency dependent impedance and reflection coefficients.

Method: Starting with the RLGC equivalent circuit, we derive the differential equations.

tjeVtv 0

dzzvLdzjziRdzzizv

dzLjRzizvdzzv

LjRzidz

dv

L

C

R

G+

v(z)

-

i(z)+

v(z+dz)

-

i(z+dz)

dzzidz

Cj

zv

dzG

zvzi

11

dzCjGzvzidzzi

CjGzvdz

di

KVL

Rearrange

Differentiate w.r.t. z

[5.4.1]

[5.4.2]

[5.4.3]

KCL

Rearrange

Differentiate w.r.t. z

[5.4.4]

[5.4.5]

[5.4.6]


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Wave Equation Revisited #2

Use the equations on the previous page to get:

tjeVtv 0

L

C

R

G+

v(z)

-

i(z)+

v(z+dz)

-

i(z+dz)

zvzvCjGLjRdz

vd 22

2

ziziCjGLjRdz

id 22

2

zR

zF eVeVzv

zR

zF eIeIzi

[5.4.7]

[5.4.8]

Which have solutions:

[5.4.9]

[5.4.10]

where jCjGLjR

[5.4.11]


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Wave Equation Solution

zRz

F eVeVLjR

CjGLjRzi

zRz

Fz

Rz

F eVeV

YZ

eVeVLjR

CjGzi

1

YZ

VI FF

YZ

VI RR

ziLjReVeVdz

dv zR

zF

Let’s work with [5.4.3] and [5.4.10] to relate the currents and voltages:

zRz

F eVeVLjR

zi

[5.4.12]

[5.4.13]

Differentiate w.r.t. z:

Substitute

[5.4.14]

[5.4.15]

[5.4.16b][5.4.16a]CjGY LjRZ

[5.4.17b][5.4.17a]

Algebra•••

where

note YZZo

so0Z

VI FF

0Z

VI RR

[5.4.19b][5.4.19a]

[5.4.18]

zRz

F eVeVZ

zi

0

1[5.4.20]

Howard Heck

IA=IFIB=IR


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Including Frequency Dependence

If a sinusoid is injected onto a transmission line, the resulting voltage can be expressed as a function of the distance from the load (z) and time.

tjzR

tjzF eeVeeVtzV ),(

Notice: The first term represents the forward traveling wave (toward

the load) The second term represents the backward traveling wave

reflected from the load (toward the source) The position dependent exponent is positive for the second

term because the wave is traveling back toward the source.

RS

v=V0ejwt

z=0

VF

VB

IF

IB

ZL

[5.4.21]


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Frequency Dependence #2 Note that and use to get:)sin()cos( je j j

tjzjR

tjzjF eeVeeVtzv )()(),(

)(sin)(cos)(sin)(cos),(

z

tjz

tVez

tjz

tVetzv Rz

Fz

[5.4.22]

[5.4.24]

[5.4.23] ztjzR

ztjzF eeVeeVtzv ),(

Separating the real and imaginary terms:

Expressing in terms of sine/cosine functions:

Where is the amplitude loss of the sinusoid

is the phase shift


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Frequency Dependence #3

zjR

zjF

tj

eVeVZ

etzi

0

,

Apply the sinusoid source to the expression for current:

ztjzR

ztjzF eeVeeV

Ztzi

0

1,

[5.4.25]

[5.4.26]


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Load Impedance

SSRF iRVVVzv 0

RS

VS

z=0

VF

VB

IF

IB

Lz

Rz

F

tj

zR

zF

tj

ZeVeV

Ze

eVeVe

tlzi

tlzv

0

,

,

zR

zF

zR

zF

L eVeV

eVeVZZ

0

[5.4.28]

[5.4.27]

[5.4.29]

Look at the boundary case.


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Reflection Coefficients & Impedance

l

F

Rl

Ftj

lR

tj

V eV

V

eVe

eVe

zv

zv

2

Vl

F

Rl

Ftj

lR

tj

I eV

V

eIe

eIe

2

F

lRF

VV V

eVV

2

1

F

lRF

II V

eVV

2

1

z

F

R

z

F

R

zR

zF

zR

zF

eVV

eVV

ZeVeV

eVeVZ

tzi

tzvzZ

2

2

00

1

1

,

,

zlv

zlv

ee

eeZzZ

22

22

0 1

1

Define the reflection coefficients:

[5.4.30]

[5.4.33]

[5.4.34]

[5.4.32]

[5.4.31]

Define the impedance in terms of reflection coefficients:

Note: most microwave texts use the gamma () symbol to represent the reflection coefficient. I have chosen to continue to use in order to remain consistent with our definition from module 2.


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Input Impedance

02

02

0 1

10

ee

eeZzZZ

lv

lv

in

v

vll

v

llv

L Zee

eeZlzZZ

1

1

1

1022

22

0

Define the input impedance:

[5.4.35]

[5.4.37]

The impedance at the load is:

lv

lv

in e

eZZ

2

2

0 1

1

Solving [5.4.36] for v, we get the familiar equation for the reflection coefficient at the load:

[5.4.38]

Substituting [5.4.37] into [5.4.30], we get the equation reflection coefficient as a function of position along the line:

[5.4.36]

zl

L

Lv e

ZZ

ZZz

2

0

0

0

0

ZZ

ZZlz

L

Lv


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Input Impedance #2

lLL

lLL

l

L

L

l

L

L

in eZZZZ

eZZZZZ

eZZZZ

eZZZZ

ZZ

200

200

02

0

0

2

0

0

0

1

1

Substituting [5.4.38] into [5.4.35] and doing the algebra:

[5.4.39]

02

2

2

2

0

020

2

20

2

0

11

11

11

11

Zee

Z

ee

ZZ

ZeZeZ

eZeZZZ

l

l

L

l

l

L

llL

llL

in

x

x

e

ex

2

2

1

1tanh

0

00 tanh

tanh

ZlZ

lZZZZ

L

Lin

Use the following relationship:

To get:


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Input Impedance #3 Alternate expression (for lossless lines):

[5.4.40]

jxjx

jxjx

eexj

eex

sin

cos Use the following relationships:

To get:

0

00

10

0Z

ee

ee

eeZV

eeV

eVeVZ

eVeV

zi

zvZ

ljV

lj

ljV

lj

ljV

ljF

ljV

ljF

lR

lF

lR

lF

in

lj

Llj

L

ljL

ljL

lj

L

Llj

lj

L

Llj

in eZZeZZ

eZZeZZZ

eZZ

ZZe

eZZ

ZZe

Z

00

000

0

0

0

0

ljljljljL

ljljljljL

in eeZeeZ

eeZeeZZ

0

0

lZljZ

ljZlZZ

L

Lin

cossin

sincos

0

0

ljZZ

ljZZZZ

L

Lin

tan

tan

0

00


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Example

118

110

566102

108.12

mradj

sm

sradj

vjj

p

LosslessLossless0

l1 = 1.5 mm

Z01 = 20 ZL = 20Z02 = 30

l2 = 2 mm

vpl = vp2 = 2 x 108 m/s

f = 18 GHz

0

00 tanh

tanh

ZlZ

lZZZZ

L

Lin

radjmmmmmmradjljl 132.1102566 13122

Looking into Z02:

j11.778+36.686

30132.1tanh20

132.1tanh3020302 j

jZ in

Use Zin2 as the load impedance to get the input impedance looking into Z01:

j14.617-18.897

20848.0tanhj11.778+36.686

848.0tanh20j11.778+36.686201 j

jZ in

radjmmmmmmradjljl 848.0105.1566 13111


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Example #2 What is v as measured at z = 0 for

the lossless transmission line system as a function of frequency?

Start with [5.4.38]:

LClj

L

Llj

L

Ll

L

Lv e

ZZ

ZZe

ZZ

ZZe

ZZ

ZZ 2

0

02

0

02

0

0

Which can be rewritten:

LCfljLCflZZ

ZZ

L

Lv 4sin4cos

0

0

Notice that the real part is zero when . Solving for f:2

4 nLCfl

LCl

nf8

where 5,3,1n

The imaginary part is zero when . Solving for f: nLCfl 4

LCl

nf

4 where 5,3,1n

ZL = 50Z0= 75

l=5 in

vp= 205 ps/ft

z=0 z=l

fjfv sec10341.5sinsec10341.5cos2.0 99


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Example #2 (2)

ZL = 50Z0= 75

l=5 in

vp= 205 ps/ft

z=0 z=l

-0.25

-0.20

-0.15

-0.10

-0.05

0.00

0.05

0.10

0.15

0.20

0.25

0.0E

+00

5.0E

+08

1.0E

+09

1.5E

+09

2.0E

+09

2.5E

+09

3.0E

+09

3.5E

+09

4.0E

+09

4.5E

+09

5.0E

+09

frequency [Hz]

Ref

lect

ion

Co

effi

cien

t

rho(real)rho(imag)


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Summary

We now have the basis for using measurement equipment to characterize interconnect in the frequency domain.


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References R.E. Matick, Transmission Lines for Digital and

Communication Networks, IEEE Press, 1995. D.M. Posar, Microwave Engineering, John Wiley

& Sons, Inc. (Wiley Interscience), 1998, 2nd edition. B. Young, Digital Signal Integrity, Prentice-Hall

PTR, 2001, 1st edition. W. Dally and J. Poulton, Digital Systems

Engineering, Cambridge University Press, 1998. Ramo, Whinnery, and Van Duzer, Fields and

Waves in Communication Electronics, 1985. U. Inan, A. Inan, Engineering Electromagnetics,

Addison Wesley, 1999, 1st edition. Ramo, Whinnery, and Van Duzer, Fields and

Waves in Communication Electronics, 1985.

Documents

© H. Heck 2008Section 5.41 Module 5:Advanced Transmission Lines Topic 4: Frequency Domain Analysis OGI ECE564 Howard Heck